 ## Formative Assessment and Bridging activities These materials are part of an iterative design process and will continue to be refined during the 2021-2022 school year. Feedback is being accepted at the link below.
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The Bridging Standards in bold below are currently live. Others are coming soon!

Standard 4.3c Standard 4.3d

Standard 4.5a

Standard 4.5b

Standard 4.5c

Standard 4.6a

Standard 4.6b Standard 4.8a

Standard 4.8b

Standard 4.8d

Standard 4.10a

Standard 4.14b

Standard 4.15

Standard 4.16

## Standard 4.2a

Standard 4.2a Compare and order fractions and mixed numbers.

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Understanding the Learning Progression

Big Ideas:

• Initial fraction ideas begin with creating experiences with a variety of visual representations (area models, number lines, set models, etc.) which can be utilized to compare and order fractions. Over time, reasoning strategies and generalizations are developed through modeling and mental imagery. There may be a transition from creating models to applying those generalizations and reasoning strategies to solve problems.

• A variety of reasoning strategies can be applied to compare fractions. Potential strategies include but are not limited to:

• reasoning about relative magnitude ;

• unit fraction reasoning;

• benchmark reasoning ( i.e. I know 4/8 is ½ so ⅝ is a little more); and

• equivalence.

• Behr and Post (1992) indicate that “a child’s understanding of the ordering of two fractions needs to be based on an understanding of the ordering of unit fractions” (1992, p.21).

• The reasoning strategy applied is often impacted by number choice. For example, I might use the benchmark of ½ if I am comparing 3/10 (a little less than ½) and ⅝ (a little more than ½).

Important Assessment Look-fors:

• Students recognize and utilize benchmark fractions.

• Students use their understanding of the ordering of unit fractions to ordering of two fractions.

• Students recognize and apply their understanding of equivalent fractions.

• Students recognize fractions equivalent to ½.

• Students represent and work with fractions greater than 1.

• Students organize their thinking in a way that helps them make sense of the problem. Student representations support their thinking/reasoning.

• Student strategies make sense for the given set of numbers.

• It is evident in the strategy used that the student understands the relationship between the numerator and the denominator.

Purposeful questions:

• Can you share where you started and why?

• How might you use your understanding of unit fractions to compare and order these fractions?

• How can benchmark be used to help you understand the value of these fractions?

• Is that fraction greater than 1? Less than 1? How do you know?

• What do you know about the value of these numbers?

• What relationships do you see?

• I’m looking at your number line and I’m noticing _____. Tell me more about that... ### Student Strengths

Students can name, write, represent and compare fractions and mixed numbers represented by a model

### Bridging Concepts

Students can make generalizations about fractional relationships across representations (i.e., when the numerator is half of the denominator, the fraction is always equal to ½).

### Standard 4.2a

Students can compare and order fractions and mixed numbers with and without models     ## Standard 4.2b

Standard 4.2b Represent equivalent fractions.

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Understanding the Learning Progression

Big Ideas:

• When two fractions are equivalent that means there are two ways of describing the same amount by using different sized fractional parts. (Van de Walle et al, 2019)

• A variety of representations and models can be used to identify different names for equivalent fractions: region/area, set and measurement models. (Students should use area representations, strips of paper, tape diagrams, number lines, counters and other manipulatives to reason about equivalence.)

• Intuitive methods using drawings and manipulatives support student understanding. Students can develop an understanding of equivalent fractions and also develop from that understanding a conceptually based algorithm. Delay sharing “a rule.” (Van de Walle et al, 2019)

Important Assessment Look-fors:

• Evidence that the student is able to identify equivalent fractions modeled using a variety of different models: area, set and/or measurement.

• The student uses strategies like removing lines or adding additional lines to the area and number line models to show how the models are equivalent.

• The student is able to use a variety of strategies and can justify their reasoning as to why fractions are equivalent.

• When the student represents their fraction as models, they represent the whole using the same size and shape model. Click this link for more information about student representations.

Purposeful questions:

• What strategy (or strategies) did you use to determine which fraction models are equivalent?

• Which fraction models are the easiest for you to identify? What made it easy?

• Which fraction models are the hardest for you to identify? What made it difficult?

• What relationships do you see?

• Can you create another equivalent model that is different from ones shown? ### Student Strengths

Students can name and write fractions and mixed numbers represented by a model. Students can represent fractions and mixed numbers with models and symbols.

### Bridging Concepts

Students can create a model to represent a fraction. (Area models tend to be easier for students to grasp while set and measurement models tend to be more difficult. Students may need additional support bridging their understanding of area models to help support their understanding of set and measurement models.)

### Standard 4.2b

Students can represent equivalent fractions.    ## Standard 4.2C

Standard 4.2c Identify the division statement that represents a fraction, with models and in context.

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Understanding the Learning Progression

Big Ideas:

• Fractions have multiple meanings and interpretations. Generally there are five main interpretations: fractions as parts of wholes or parts of sets; fractions as the result of dividing two numbers; fractions as the ratio of two quantities; fractions as operators; and fractions as measures (Behr, Harel, Post, and Lesh 1992; Kieren 1988; Lamon 1999). When a fraction is presented in symbolic form, devoid of context, the intended interpretation of the fraction is not evident. The various interpretations are needed, however, in order to make sense of fraction problems and situations. Students need to explore and understand that fractional parts are equal shares of a whole or set model.

• A fraction can also represent the result obtained when two numbers are divided. This interpretation of fraction is sometimes referred to as the quotient meaning, since the quotient is the answer to a division problem. Chapin and Johnson (2000) give these examples, “the number of gumdrops each child receives when 40 gumdrops are shared among 5 children can be expressed as 40/5 , 8/1 , or 8; when two steaks are shared equally among three people, each person gets 2/ 3 of a steak for dinner. We often express the quotient as a mixed number rather than an improper fraction – 15 feet of rope can be divided to make two jump ropes, each 7 1/ 2 ( 15/2 ) feet long” (p .99– 101).

• When partitioning a whole into more equal shares the parts become smaller. (Teaching Student-Centered Mathematics, Grades 3-5, John Van de Walle)

• When exploring the concept of fractions and connecting it to the division statement, students should be able to identify and recognize that the fraction is the amount that each person would receive when dividing equally.

Important Assessment Look-fors:

• The student is able to accurately partition models when exploring fair share problems.

• When given context, the student is able to successfully use models to identify the division statement that represents the fraction.

• The student can interpret the fraction and division statement as the amount each person would receive.

• The student can identify the difference between a division statement that represents an improper fraction versus a proper fraction within context. The student can also represent a fair share problem when the numerator is larger or smaller than the denominator.

Purposeful questions:

• Will each person receive more or less than a whole? Explain your reasoning.

• Identify the division statements that represent this fraction as presented in the context.

• How much will each person receive when shared equally?

• Once the students have discovered how much each person will receive, ask the students if each person was to combine their equal share together, what would be the combined total? What do you notice?

• Example: 2 cookies shared with 3 students. Each person would receive ⅔ of a cookie. When combining the equal shares from each of the 3 people, ⅔ + ⅔ + ⅔ = 6/3 or 2. The sum is equivalent to 2 whole cookies or the original amount of cookies that was shared.

• If the dividend is smaller than the divisor, how does this relate to how much each person would receive if the amount was shared equally? Would this be true if the dividend was larger than the divisor?

• Example: 2 pizzas shared with 3 people versus 3 pizzas shared with 2 people. ### Student Strengths

Students can name and write fractions and mixed numbers represented by a model.

### Bridging Concepts

Students understand the concept of division and know what the dividend and divisor represents.

### Standard 4.2C

Students can identify the division statement that represents a fraction, with models and in context.   ## Standard 4.3a

Standard 4.3a Read, write, represent, and identify decimals expressed through thousandths.

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Understanding the Learning Progression

Big Ideas:

• The structure of the base-ten number system is based upon a simple pattern of tens, where each place is ten times the value of the place to its right. This is known as a ten-to-one place value relationship (Van de Walle et al., 2019)

• Decimals is another form of writing fractions and the connection between the two is important in understanding the concepts of decimals (i.e. connect 1/10 as 0.1 and 1/100 as 0.01 and 1/1000 as 0.001 - Reading the decimal fractions will help students “hear” the connection).

• Understanding of the base-ten system to the relationship between adjacent places and how numbers compare can help support students round for decimals to thousandths. For example, it is important to deepen understanding and fluency with decimals in the different forms, seeing .57 as 5 tenths and 7 hundredths as well as 57 hundredths (Common Core Standards Writing Team, 2019, p. 64). This ability to rename and decompose decimals can help students round to the nearest whole number, tenth or hundredth.

• The decimal point separates the whole from the fractional part. The place value system extends infinitely in both directions of the decimal point, to very large and very small numbers. Connected uses of decimals in real life using money, metric measurements, batting averages can support student understanding.

Important Assessment Look-fors:

• The student is able to identify the decimal represented using a variety of different models.

• The student can model a given decimal using base ten blocks when the whole is defined.

• The student is able to read and write decimals, especially with zero placeholders?

• The student is able to represent decimals in a variety of forms?

• When given a decimal, the student can identify the place value position and value of each digit.

Purposeful questions:

• Is the decimal represented more or less than a whole? Explain your answer.

• If the whole changed, how would that affect the decimal represented?

• When modeling with base ten blocks, what would the cube represent if the whole is equal to a rod? Explain your answer.

• How can you represent this decimal in a variety of ways, such as number line, money, or 10-by-10 grid? ### Student Strengths

Students can identify the ten-to-one relationship within the base-ten system of whole numbers.
Students can read and write various amounts of money and recognize that the number before the decimal point represents whole dollars and the amount to the right of the decimal point represent a part of a dollar.

### Bridging Concepts

Students connect the idea that money is a model or representation of decimals (i.e., that 10 dimes equals a dollar or 100 pennies is equal to a dollar). Students can extend their understanding of ten-to-one place value relationships to decimals.

### Standard 4.3a

Read, write, represent, and identify decimals expressed through thousandths.    ## Standard 4.3c

Standard 4.3c Compare and order decimals.

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Understanding the Learning Progression

Big Ideas:

• This standard builds upon the work students did in previous grades in understanding place value and comparing and ordering whole numbers. In grade 4, in addition to comparing greater numbers, students began relating decimal fractions and decimal numbers and comparing decimals using visual models. The place value understanding that supports the ability to compare decimals also supports the understanding of rounding decimals, which is introduced in grade 5 as SOL 5.1.

• Concepts of whole numbers, fractions, and decimals are connected and applied when comparing and ordering.

• Using manipulatives to construct decimals helps students develop an understanding of the relative size of decimal numbers for comparing and ordering.

• It is important for students to connect decimal number sense concepts such as representations, decimals benchmarks, and/or fractions when comparing and ordering decimals.

Important Assessment Look-fors:

• The student can compare decimals with different amounts of digits. (Example 0.9 and 0.234)

• The student can justify which decimal is larger or smaller using a variety of strategies that focus on number sense such as models, decimal benchmarks and/or identifying the value of the greatest place value.

• The student can order decimals least to greatest or greatest to least.

• The student can apply a variety of strategies when ordering decimals with similar digits and/or different amounts of digits. (Example: 0.9; 0.901; 0.09; 0.009)

Purposeful questions:

• Identify which decimal is the greatest? Which one is the least? Explain your answer.

• Which decimal(s) can be placed in the space provided so that the decimals are in order from least to greatest? (Example: 0.142; 0.45 ______; 0.8)

• Compare the following decimals using two different strategies to justify which one is greater or least. ### Student Strengths

Students can compare and order whole numbers with similar numbers of digits and/or smaller numbers.

### Bridging Concepts

Students use understanding of the ten-to-one base ten relationships to create decimal representations (i.e., base ten blocks, decimal circles/squares, etc.).

### Standard 4.3c

Students can compare and order decimals.   ## Standard 4.4a

Standard 4.4a Demonstrate fluency with multiplication facts through 12 x 12, and corresponding division facts.

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Understanding the Learning Progression

Big Ideas:

• Computational fluency is the ability to think flexibly in order to choose appropriate strategies to solve problems accurately and efficiently (VDOE Grade 4 Curriculum Framework).

• All of the facts are conceptually related so students can figure out new or unknown facts using what they already know (Van de Walle et al, 2018).

• The development of computational fluency relies on quick access to number facts. There are patterns and relationships that exist in the facts. These relationships can be used to learn and retain the facts (VDOE Grade 4 Curriculum Framework).

• Mastering the basic facts is a developmental process. Students move through phases, starting with counting, then more efficient reasoning strategies, and eventually quick recall and mastery. Instruction must help students through these phases without rushing them to know their facts only through memorization (Van de Walle et al, 2018).

• When students struggle with developing basic fact fluency, they may need to return to foundational ideas. Just providing additional drill will not resolve their challenges and can negatively affect their confidence and success in mathematics (Van de Walle et al, 2018).

• In order to develop and use strategies to learn the multiplication facts through the twelves table, students should use concrete materials, a hundreds chart, and mental mathematics. Strategies to learn the multiplication facts include an understanding of multiples, properties of zero and one as factors, commutative property, and related facts (VDOE Grade 4 Curriculum Framework).

Important Assessment Look-fors:

• The student knows and is able to apply a variety of strategies (i.e., partial products, using friendly numbers, repeated addition, and/or decomposition strategies, recall, etc.).

• The student demonstrates an understanding of the term “product” and uses a strategy to find a product that leads to a correct answer.

• The student identifies multiple number sentences with the same product.

• Student’s work shows understanding of the inverse relationship between multiplication and division.

Purposeful questions:

• What strategies are most efficient for this fact (or set of facts) and how do you know?

• How can knowing one fact help you with another fact (or a fact that you don’t know)?

• What makes one fact related to another fact? How can knowing related facts be helpful?

• What do you know about the relationship between multiplication and division? How can that relationship help you solve problems?

• What are some ways that you can break apart these numbers to make this problem easier? ### Student Strengths

Students developed an understanding of the meanings of multiplication and division of whole numbers through activities and practical problems involving equal-sized groups, arrays, and length models.

### Bridging Concepts

Students have developed efficient reasoning strategies that will lead them to quick recall and memorization of facts of 0, 1, 2, 5 and 10.

### Standard 4.4A

The student will demonstrate fluency with multiplication facts through 12 × 12, and the corresponding division facts.     ## Standard 4.4b

Standard 4.4b Estimate and determine sums, differences, and products of whole numbers.

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Understanding the Learning Progression

Big Ideas:

• Flexible methods of computation for all four operations involve taking apart (decomposing) and combining (composing) numbers in a variety of ways (Van de Walle et al, 2018).

• Students should explore and apply the properties of addition and multiplication as strategies for solving addition, subtraction, multiplication, and division problems using a variety of representations (e.g., manipulatives, diagrams, and symbols). (VDOE Grade 4 Curriculum Framework)

• Flexible methods for computation require deep understanding of the operations and the properties of operations (commutative property, associative property, and the distributive property). How addition and subtraction, as well as multiplication and division, are related as inverse operations is also critical knowledge (Van de Walle et al, 2018).

• Estimation can be used to determine the approximation for and then to verify the reasonableness of sums, differences, products, and quotients of whole numbers. An estimate is a number that lies within a range of the exact solution, and the estimation strategy used in a particular problem determines how close the number is to the exact solution. (VDOE Grade 4 Curriculum Framework)

Important Assessment Look-fors:

• Students' work shows that they understand and can apply terms such as estimate, sum, difference and product.

• Students’ can select a strategy that they understand and can apply successfully.

• Students can explain their solution and justify the reasonableness of their answer.

• Student’s work contains a variety of strategies and representations.

• Students are able to use estimation to determine the reasonableness of their answer (i.e., a little more than, a little less than, closer to, etc.).

Purposeful questions:

• Tell me about the operation you decided to use and why it makes sense.

• How did you represent your thinking?

• How do you know __ is closest to __?

• Why did you choose that place value to estimate? ### Student Strengths

The student will estimate and determine the sum or difference of two whole numbers up to 9,999.
The student will represent multiplication and division through 10 × 10, using a variety of approaches and models.

### Bridging Concepts

Students use their place value understanding and ability to round to estimate sums and differences of two whole numbers.Students use strategies based on place value and properties of the operations to multiply whole numbers.

### Standard 4.4B

The student will estimate and determine sums, differences, and products of whole numbers.    ## Standard 4.4c

Standard 4.4c Estimate and determine quotients of whole numbers, with and without remainders.

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Understanding the Learning Progression

Big Ideas:

• Flexible methods of computation involve taking apart (decomposing) and combining (composing) numbers in a variety of ways (Van de Walle et al, 2018).

• Students should explore and apply the properties of addition and multiplication as strategies for solving division problems using a variety of representations (e.g., manipulatives, diagrams, and symbols). (VDOE Grade 4 Curriculum Framework)

• Flexible methods for computation require deep understanding of the operations and the properties of operations (commutative property, associative property, and the distributive property). How addition and subtraction, as well as multiplication and division, are related as inverse operations is also critical knowledge (Van de Walle et al, 2018).

• Estimation can be used to determine the approximation for and then to verify the reasonableness of quotients of whole numbers. An estimate is a number that lies within a range of the exact solution, and the estimation strategy used in a particular problem determines how close the number is to the exact solution (VDOE Grade 4 Curriculum Framework).

• Some division situations will produce a remainder, but the remainder will always be less than the divisor. If the remainder is greater than the divisor, that means at least one more can be given to each group (fair sharing) or at least one more group of the given size (the dividend) may be created (Georgia Department of Education Grade 4 Curriculum).

Important Assessment Look-fors:

• Students’ use methods they understand and can explain.

• Students' work shows that they understand and can apply the term quotient(s).

• Students’ work demonstrates an understanding of place value and identifying related facts that correlate with the problem.

• Students can explain their solution and the reasonableness of their answer.

• Student’s work contains a variety of strategies and representations.

• The student is able to use estimation to determine the reasonableness of their answer (ie, a little more than, a little less than, closer to, etc.).

Probing questions:

• How did you represent your thinking?

• How do you know __ is closest to __?

• Why did you choose that place value for your estimate?

• What is the meaning of a remainder in a division problem?

• What effect does a remainder have on a quotient?

• How are remainders and divisors related? ### Student Strengths

Students developed an understanding of the meanings of multiplication and division of whole numbers through activities and practical problems involving equal-sized groups, arrays, and length models.
The students have worked on fluency of facts for 0, 1, 2, 5, and 10.

### Bridging Concepts

Students may still be working on developing efficient reasoning strategies that will lead them to quick recall and memorization of facts from 0 - 12.

### Standard 4.4c

The student will estimate and determine quotients of whole numbers, with and without remainders.     ## Standard 4.4d

Standard 4.4d Create and solve single-step and multistep practical problems involving addition, subtraction, and multiplication of whole numbers and single step practical problems with division.

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Understanding the Learning Progression

Big Ideas:

• Use reasoning and a variety of strategies that the student understands and is able to explain.

• Makes sense of the problem rather than relying on keywords that may not always be helpful. (See VDOE Standards of Learning- Grade 4 Document p.19).

• Inverse operations are related and can flexibly be used to solve problems.

• Estimation, based on number sense and an understanding of place value, can be used to determine if an answer is reasonable.

• Students will be stronger problem solvers given opportunities to engage with a variety of problem types. For more information about addition/subtraction problem types see the Grade 3 VDOE Standards of Learning Document p. 15 and for multiplication/division problem types see the Grade 4 VDOE Standards of Learning Document pp.20-21.

Important Assessment Look-fors:

• Students are able to create a problem using the given information.

• Students are able to correctly represent the problem they created and it matches the way they solved it.

• Student work shows that they understand the problem because they have planned an approach to solve the problem and/or a way to represent their thinking.

• Student work shows that they understand what each number in the problem represents.

• Student work shows that they understand what operations can be used, the meaning behind the operation or how the operations are related.

Purposeful questions:

• How did you represent your thinking? How do you know it matches the story?

• Is there another way that you could solve this problem?

• What equation/number sentence matches your thinking?

• What operations did you decide to use? Why? ### Student Strengths

Students determine the sum or difference of two whole numbers to 4 digits.
Students create and solve single-step and multistep practical problems involving sums or differences of two whole numbers, each 9,999 or less.

### Bridging Concepts

Students use their understanding of multiplication and division to solve a variety of single step contextual problems.
Students use place value understanding and properties of operations to solve multiplication problems up to 10 x 10.

### Standard 4.4D

Students create and solve single-step and multistep practical problems involving addition, subtraction, and multiplication of whole numbers and single step practical problems with division.      ### Routines: 